Since 2005 it is one of the most popular games in the world, especially under the beach umbrella. We are talking about the **Sudoku**the famous puzzle composed of a **9×9 grid** divided into **9 blocks ****3×3**with one fundamental rule: **Enter the numbers 1 through 9 so that they appear once and only once in each row, column, or block.**

But how do you solve them? Is there a method that, if used, will always lead us to complete them successfully?

## Sudoku Math and Solving Strategies

Sudoku is now one of the most popular puzzle games in the world. Its **invention** official dates back to **1979** in the USA, even though at the time it was called ** Numbers in Place**. His consecration to

*instead, as you can imagine, it arrived in Japan in*

**Sudoku****1984,**when he was renamed with his iconic name and his popularity then exploded in 2005, until he became the “king of summer”.

The underlying concept is simple: fill the 9×9 grid

**by entering the numbers 1 through 9 so that they appear once and only once in each row, column, or block**

**3×3**In this case we say that the sudoku is

**order 9.**

There are also variants of greater or lesser order, but the classic version is the one with 9.

Anyone who has played it at least once has asked themselves: **Is there a mathematical method that allows us to solve it with certainty?**

The answer is neither. The mathematical rule exists, and it is precisely the unique rule that we mentioned earlier. It is precisely this **rule** which, if the right conditions exist, will lead us to the resolution of the puzzle thanks to the constraints it creates on each box. Of course, putting it into practice is not easy and precise conditions are necessary. For example, it is necessary that the **Initial clues** are **no less than 17**. If there were 16, in fact, there is no mathematical certainty that the solution is one and only one. Then, of course, the clues must be positioned by the creator of the puzzle so that they provide us with the necessary constraints to arrive at a solution.

That said, the question now is: **How can we exploit the single rule to arrive at a solution?** There are several **techniques,** some more basic, some more complex.

## The “forced choices” given by the clues

First of all, we look for the **“forced choices”**that is, the values that are clearly indicated to us by the numbers already present in the grid. There are two ways:

**I focus on a number**that I want to insert into a block (or row or column) and I see how the constraints given by the columns or rows impose a certain position on me.

**I focus on a column (or row or block)**and depending on the missing numbers in the column and the constraints given by rows and blocks, I complete the column.

## The importance of notes in resolution

The examples given so far are certainly known to most. We now come to the fundamental strategy for solving these puzzles: **mark all the possibilities** in each box and gradually discard them. I know, it may seem boring and confusing, but done methodically it is in fact the only (or almost, we’ll get there) tool at our disposal.

The “trick” is therefore to start from the clues at your disposal and, through the only rule we have, begin to **write down the possible numbers** in each cell, going street by street **excluding options that create contradictions**. There are some techniques that make this process easier:

**identify the obvious singles**: once all the notes have been written, we look for the cells in which the possibilities are unique and therefore forced.**identify obvious pairs or trios:**when in two boxes (in the case of pairs, three boxes in the case of three of a kind) only two numbers are possible, this means that they will necessarily be there, because if they were elsewhere, no other number could fill those boxes. This allows us to eliminate the two numbers (or three) from the options of any other box in which they could appear.**locate hidden singles:**When within a block, row or column we find a number that appears in the possibilities of only one box, then we can erase all the other possibilities and insert that number.**locate hidden pairs or sets:**When more than one number is possible in two boxes (in the case of pairs, or three boxes in the case of tic-tac-toes), but two (or three) specific numbers can only appear in those cells, this means that the other options can be eliminated.**identify pairs or trios by exclusion:**It may happen that within a block, a number is possible along only one row or column. In this case, we can delete this number from the possibilities that appear along the same row or column within other blocks.

## Trial and error is a great strategy

Another fundamental tip is: **go by trial and error**. It may seem absurd, but trying to put your intuitions into practice, even if we have no certainty that it is true, can lead to **important discoveries**. For example, in the cell highlighted in the image below, the only possibilities are the **3** and the **7**.

We can think of **try with the choice of number ****3** and two scenarios can arise: the choice **turns out to be a winner**that is, we do not run into contradictions, or the choice **it turns out wrong**that is, we arrive at a point where some numbers will be repeated on the same row, column or block. If the first case were to occur, our intuition was right. If the second were to occur, we can delete all the numbers inserted after the 3 and at this point be **certain** that – if 3 is not the right choice – it **It will definitely be the ****7**!